# Force of magnetic attraction between hemispheres

Pouyan

## Homework Statement

Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω and surface charge density σ.

## Homework Equations

Maxwell Tensor : Tm = [/B](1/μ) * ((B*n)B - (1/2)*B2 n)

B_in =
(2/3)*μσRω*z
B_out =
μm/(4πr3) * (2cosθ r + sinθ θ)
where
m = (4/3)*πR3(σωR)

## The Attempt at a Solution

I do see in my solution that :
1-We use a surface consisting of the entire equatorial plane, CLOSING IT with a hemispherical surface at infinity where (since the field is zero out there) contribution is zero.
for r>R :
B = μm/(4πr3) θ = - μm/(4πr3) z (Why ?! )
2-
We have a equatorial circular disk. We use B inside and da = rdrdφ. Direction is in z-axis!
I don't want the solution because I already have it but MY QUESTIONS are :

A) How should think and imagine this spheres ?!
Should I think like this and use those formulas : B) Why we do write B = μm/(4πr3) θ = - μm/(4πr3) z
Why is θ =-z in this case ?! I don't get it !

I know the rest. That is just integrating

#### Attachments

Homework Helper
Gold Member
A) How should think and imagine this spheres ?!
Should I think like this and use those formulas :
I'm not quite following what you are asking here. Can you reword the question?

B) Why we do write B = μm/(4πr3) θ = - μm/(4πr3) z
Why is θ =-z in this case ?! I don't get it !

It is because you are using spherical coordinates with θ measured from the positive z axis. The unit vector θ points in the direction of increasing θ. For a point on the equatorial plane of the sphere, what is the direction of θ?

• Pouyan
Pouyan
I'm not quite following what you are asking here. Can you reword the question?

It is because you are using spherical coordinates with θ measured from the positive z axis. The unit vector θ points in the direction of increasing θ. For a point on the equatorial plane of the sphere, what is the direction of θ?
I've got it !

Thanks